Scrub and wash the mussels removing any hairy bit poking out by pulling on it gently. Don't force them open - they will open as they cook in the broth
Chop the onion and garlic into small pieces
Add the oil to the frypan and heat without letting it get too hot and smoky
Add the chopped onion and garlic to the pan and saute for 2 or so minutes. They can both brown a little but don't overdo it
Put the onion, garlic and the canned tomato into the blender. Pulse the mixture till you have a smoothish puree
Meanwhile, slice the carrot thinly
Add the carrot and peas to the frypan. Add a cupful of water or more if the puree is very thick. You want it to be thick enough to have 'body' but not so thin that it is watery
Cook the broth at a low boil for 5 mins or so till the carrot is tender - you might like to test a slice, see if it is as tender as you like and if not cook it a little longer
Add the two tbsp of cream to the broth it through well.
Add the pepper and salt to the broth, mix through and taste to see if it's salty enough for you. The mussels will add some salt as they cook so don't oversalt at this stage.
Add the mussels in their shell to the broth. The broth doesn't have to cover the mussels, but check that it isn't getting too thick. Add a cupful of water if it is.
Let the broth and mussels cook till the mussels open. They won't open all at the same time - that's okay, you are unlikely to overcook them. Also, some may only open a little while others open wide - that's fine also.
Chop the coriander roughly
When all the mussels are open - the timing for this cannot be given precisely but it will be no more than 10 minutes usually - scoop them out into a bowl in preparation for assembling and serving the soup
Spoon the soup into soup bowls, add 8 - 10 mussels per person into the bowl, sprinkle over with the chopped coriander